18.904 — Seminar in Topology — Spring 2011

Description

This course is a seminar in topology. The main mathematical goal is to learn about the fundamental
group, homology and cohomology. The main non-mathematical goal is to obtain experience giving
math talks. Lectures will be delivered by the students, with two students speaking
at each class. There are no exams. There will be some homework assignments and a final paper.

Seminar leader

Time and location

The seminar typically meets Monday, Wednesday and Friday from 12pm to 1pm in room 2-139. See the calendar
for exceptions. Practice lectures will also take place in room 2-139.

Textbooks

We will mainly use Hatcher's “Algebraic topology.” This book is available for free online at
Hatcher's webpage. (It is also available in print.) We
may also make some use of Massey's “A basic course in algebraic topology,” which is published by
Springer in the Graduate Texts in Math series (GTM 127).

Grading

The final grade is determined as follows:

60% — Lectures and participation

30% — Final paper

10% — Problem sets

Attendance is mandatory. Every three missed classes will result in the drop of a letter grade; thus one can miss up
to two classes with no effect on the grade. The classes on 2/2 and 2/4 will not count towards this. Classes missed
for a valid medical excuse will also not count towards this. If you know you will miss a class for some reason,
e-mail me a day or two in advance and we can try to work something out.

Lectures and participation

Each class two students will give lectures. Each lecture should be about 25 minutes long. Individual lectures will
not be graded, but lectures make up a good portion of the final grade. In evaluating your lectures, I will look at
their clarity, organization and preparedness. I will also consider how your lectures improve over the course of the
semester.

You will give a practice lecture to a small audience (consisting of me, Susan Ruff and the other student lecturing
in the same class as you) before your first lecture.

Each lecturer will give one or two exercises relevant to the material being presented. These exercises,
and their solutions, should be e-mailed to me as a Latex file. The exercises can be stated during lecture, though
this is not necessary. It's ok if the exercises come from a book (although it'd be preferable if they did not, or
at least if they were slightly modified), but be sure to give proper attribution.

As a member of the audience, I'd like you to write a few comments on each lecture you observe. I'm not asking for
any kind of lengthy analysis; it would be enough to point out that the lecturer is writing too small. However,
make sure the comments are useful — don't just say “that proof was good,” say why. I will collect
these comments at the end of class and e-mail them to the lecturer so that they can have some feedback. (The
lecturer will not know who made which comments.)

Homework

There will be approximately four problem sets. These will count towards the final grade. Solutions are to be
written in Latex. You may work together on the problem sets, but everyone must write up their own solutions.

There will also be exercise sets, mainly composed of exercises given by lecturers. These are optional and do not have
to be turned in. If you are intested in learning the material, it is probably a good idea to do at least some of the
exercises.

Final paper

The final paper is an exposition of a topic in algebraic topology that we will not cover in the seminar. It must be
at least 10 pages long and written in Latex. Topics will be selected for the papers in March. A first draft is
due in April, and a final draft two weeks later. In the final six or so meetings of class, students will give talks
on their final papers.

Give the idea of the fundamental group and some basic examples. Also
give the idea of homotopy equivalences and some basic examples.

Define what a path in a space is, and what a homotopy of paths is. Show
that “homotopic” is an equivalence relation on paths. Give an example of a
homotopy between two paths. Give an example of two paths with the same endpoints which
are not homotopic (proof not required). Define the composition of two paths, and
show that it is well-defined on homotopy classes of paths.
References: Hatcher §1.1, Massey §II.2.

Define the fundamental group of a topological space and prove that it is a group.
Give an example of a space whose fundamental group is trivial and an
example of a space whose fundamental group is non-trivial (without proofs). Show that
a path between two points induces an isomorphism between the fundamental groups based
at those points.
References: Hatcher §1.1, Massey §II.3.

Define a homomorphism Z → π1(S1) and show that it is
an isomorphism.
References: Hatcher §1.1, Massey §II.5.

Using the isomorphism π1(S1) = Z from the previous lecture,
prove the following theorems: the fundamental theorem of algebra (every non-constant
polynomial with complex coefficients has a complex root), Brouwer's fixed point
theorem for the 2-disc (every continuous map from the 2-disc to itself has a fixed point)
and the Borsuk–Ulam theorem for the 2-sphere (every continuous map from the 2-sphere
to R2 admits two antipodal points at which it is equal).
References: Hatcher §1.1, Massey §II.6.

Define what it means for a space to be contractible and simply connected.
Prove that a contractible space is simply connected. Show that Rn is
contractible. Show that a sphere of dimension at least 2 is simply connected. (Spheres
are not contractible though!)
References: ?

Show that the fundamental group of the cartesian product of two spaces is the direct product
of the fundamental groups of the two spaces. Compute the fundamental groups of tori.
(A torus is just a product of circles.)
References: Hatcher §1.1, Massey §II.7.

Show that a continuous map f : X → Y of topological spaces induces a homomorphism of
groups f* : π1(X) → π1(Y). Define what it means
for two maps f,g : X → Y to be homotopic (generalizing the definition for paths).
Show that if f and g are homotopic then f*=g*.
References: Hatcher §1.1, Massey §II.4.

Define what a homotopy equivalence between two spaces is. Show that if two spaces
are homotopy equivalent then their fundamental groups are isomorphic. Give several examples
of spaces which are and are not homotopy equivalent.
References: Hatcher §1.1, Massey §II.8.

Define the free group on n letters Fn. Define the wedge sum
X ∨ Y of two topological spaces. Construct a map F2 → S1 ∨
S1 and show that it is an isomorphism. State a similar result (without proof)
for an n-fold wedge sum of circles.

Define the amalgamated free product of groups. Prove the universal property
of this product. Give some examples.

State van Kampen's theorem. Prove the surjectivity part of the statement.

Prove the injectivity part of van Kampen's theorem.

Define what a covering space is. Give some ideas about the universal
cover and the Galois correspondence for covers. Do some examples.

Construct the universal cover.
References: Hatcher pp. 63–65.

Verify properties of the construction of the universal cover from the
previous lecture.
References: Hatcher pp. 63–65.

Prove the homotopy lifting property. Deduce the path lifting property
as a corollary. Show that p* is injective on fundamental
groups if p is a covering map.
References: Hatcher Prop. 1.30 and Prop. 1.31.

Prove the criterion for lifting and uniqueness of lifts.
References: Hatcher Prop. 1.33 and Prop. 1.34.

Show that for each subgroup of the fundamental group, there exists a
corresponding cover. Show that this cover is Galois if and only if
the subgroup of the fundamental group is normal.
References: Hatcher Prop. 1.36 and Prop. 1.39.

Finish off the proof of the Galois correspondence.
References: Hatcher Prop. 1.37 and Thm. 1.38.

Discuss the theory of quotients by discontinuous group actions, in particular,
how the fundamental group changes. Compute the fundamental group of
real projective space.
References: Hatcher Prop. 1.40.

An overview of what we'll cover on homology.

Define what a singular simplex is. Define the group of singular
chains and the boundary operator on it. Prove that the square
of the boundary operator is 0.
References: Hatcher p. 103 and p. 108.

Define the group of cycles and boundaries. Define
singular homology. Compute H0 and the homology of a point.
References: Hatcher p. 108, Prop. 2.7 and Prop. 2.8.

Define what a chain complex is. Define what a map of chain
complexes is. Define what a homotopy of maps of chain complexes is.
Show that a map of chain complexes induces a map on homology, and that
homotopic maps induce the same map on homology. This talk is pure algebra,
and does not involve topology in any way.
References: Some discussion in Hatcher pp. 110–113.

Show that homology is naturally a functor on the category of topological
spaces. Show that homotopic maps induce the same map on homology.
References: Hatcher pp. 110–113, esp. Thm. 2.10.

Show that there is a natural long exact sequence in homology associated to
a short exact sequence of chain complexes. This talk is pure algebra, and
does not involve topology in any way.
References: Some discussion in Hatcher pp. 115–118.

Define the relative homology groups. Show that there is a long exact
sequence relating the homology of a space, a subspace and the relative
homology groups. Do the example of a disc relative to its boundary.
References: Hatcher pp. 115–118, esp. Example 2.17.

State the theorem of excision. State Proposition 2.21 of Hatcher,
and prove the excision theorem assuming this proposition.
References: Hatcher p. 119 (esp. Thm. 2.20) and p. 124.

Introduce the cellular chain complex associated to a CW complex. State
Theorem 2.35, that the homology of this complex is naturally isomorphic to
singular homology. Also, mention the cellular boundary formula.
References: Hatcher pp. 137–141.

Do some examples of cellular homology, for instance, recompute the homology
of the n-sphere, compute the homology of complex projective space.
References: Hatcher pp. 141–146.

Discuss the functor Hom(-, G). Define the cohomology of a space, with
coefficients in G.
References: Hatcher pp. 197–198.

Mention the formal properties of cohomology, which parallel those of homology.
Be sure to point out the places where there are small differences (e.g., when
the arrows go the other way). Also state the universal coefficient theorem,
which let's one compute cohomology from homology.
References: Hatcher pp. 199–204.

Define the cup product, say that it makes cohomology into a ring. Show that the
map on cohomology induced from a map on spaces is a ring homomorphism. Finally,
state and try to prove Theorem 3.14, which says that the cup product is graded
commutative.
References: Hatcher pp. 206–207, Prop. 3.10, Thm. 3.14.

Compute the cohomology ring of the wedge sum S2 ∨ S4 and
of complex projective space CP2. Observe that the underlying
groups are isomorphic, but since the ring structure is different the two spaces
are not homotopy equivalent. This is something that cannot be seen purely in terms
of homology, and relies on the additional structure in cohomology.
References: Hatcher Example 3.13.

Define the cross product in cohomology. State the Kunneth formula in
the case
where cohomology is finitely generated and free, and in the general case. Do
an example, e.g., compute the cohomology of a torus.
References: Hatcher pp. 218–219, Appendix 3.B.

Sketch a proof of the Kunneth formula in the simple case where the cohomology is
finitely generated and free.
References: Hatcher pp. 220–221.

Review the definition of a manifold. State a primitive version of Poincare
duality. Explain what this means in the case of a torus.
References: Hatcher p. 231.

Define what a local orientation is and what a (global) orientation
is. Discuss
the two sheeted cover which determines if there is an orientation. Define what a
section is, in this context.
References: Hatcher pp. 233–235.

State Theorem 3.26 and define the term fundamental class. State and
prove Lemma 3.27.
References: Hatcher p. 236.

Instructions for problem sets

You may work together, but everybody must write-up their own solutions.

Solutions may use results from class, but should otherwise be self-contained.
You may freely use basic results without proof or reference.

Feel free to consult books to review material needed to complete the problems, but don't
look up solutions to the problems!

Solutions must be written in Latex, printed, stapled and handed-in at the beginning of class.
Please make your solutions readable; feel free to state intermediate results as lemmas. It
may be cleanest to make each problem its own section. Also, set the margins to something
reasonable (the default margins in Latex are not reasonable).

Each problem will be graded out of 10 points, unless otherwise indicated.

Problem sets

Lecture notes

For one round of lectures, the students submitted typed up lecture notes.
We decided not to continue this, as it took too much time and books
already have the same material.

Exercises

For a while, students composed exercises related to the material on which
they were lecturing. These exercises were intended as practice for the
other students with the material, and were not required to be completed.

Concering the final paper

This document gives a list of mistakes that were
common in the drafts of your final papers. Every draft that I read
contained at least one of these mistakes, so I recommend that you all
look through this and apply what it says to your writing.

A sample file

This note (and its tex source) gives some examples of how to do things
like matrices and commutative diagrams, and also discusses some things
that you should and should not do in latex (repeated in abridged form
below).

Last updated: 3/3/11.

Stuff not to do

These remarks are elaborated upon in the above file, but put here for
convenience and emphasis:

Do not begin sentences (or phrases) with math. Do not put math next to
math.

Multi-letter operators and functions, like sin, should not be
italicized.

Use the correct size parthenses.

When multiple formulas are put into a single displaymath, put space
between them.

The default margins are much too large; adjust them.

The final paper is an approximately 10 page exposition on a topic in algebraic topology not covered in our seminar.
The paper must be written in Latex (or some other flavor of Tex).

You must select the topic for your paper by March 7th. I'd prefer that no two of you do the same topic, so if
there's something you'd really like to do you should tell me soon. When you know what you want to do, just send
me an e-mail.

Below is a list of possible ideas for topics. You're free to choose something not on this list, but run it by
me first.

Concrete topics

Hopf fibrations. The Hopf fibrations are three specific maps of spheres defined using projective spaces
over the complex numbers, quaternions and octonions. They were the first non-trivial elements of higher homotopy
groups of spheres to be discovered.
This topic has been taken.

Lens spaces. The lens spaces are an explicit family of 3-dimensional manifolds indexed by pairs of
integers. They provided the first example of a pair of compact manifolds which are homotopy equivalent but not
homeomorphic.

PSL2(Z) as a free product. Using some algebraic topology, one can give a nice proof
that PSL2(Z) (roughly the group 2×2 integer matrices) is the
free product of a cyclic group of order 2 and a cyclic group of order 3.
This topic has been taken.

Homology theories

Borel–Moore homology and cohomology with supports. These variants of homology and cohomology
are useful when dealing with non-compact spaces. They allow one to extend Poincaré duality to the
setting of non-compact manifolds.

Simplicial and cellular homology. Simplicial complexes give a very elementary and combinatorial way to
describe topological spaces. Simplicial homology describes how to compute the homology of a space that is described
as a simplicial complex. CW complexes also provide a way to describe topological spaces, one that is less
elementary but more flexible than that provided by simplicial complexes. Cellular homology describes how to compute
the homology of space described as a CW complex.
This topic has been removed.

de Rham cohomology. This is a cohomology theory for smooth manifolds defined in terms of differential
forms. The main theorem, the de Rham isomorphism theorem, states that de Rham cohomology is isomorphic to singular
cohomology (with real coefficients). This is remarkable since the definition uses the smooth structure on the
manifold but the final product is only dependent on the topology. p-adic analogues of this result have been very
important in number theory in recent years.
This topic has been taken.

Sheaf theory. Sheaves are to topological spaces what modules are to rings. There is a notion of
cohomology for a sheaf on a topological space; the cohomology of the so-called “constant sheaf” gives
the singular cohomology of the space. Sheaf theory is indispensable in modern algebraic geometry.
This topic has been taken.

K-theory. Given a space X, a group K0(X) is constructed using complex vector bundles on X.
Groups Kn(X) can be constructed in various manners. The K-groups are not isomorphic to the singular
cohomology groups. K-theory forms what is called an “extraordinary cohomology theory.&rdquo The most important result
is Bott periodicity, which states that the K-groups are periodic with period 2.
This topic has been taken.

Categorical methods

Simplicial sets. The theory of simplicial sets offers a model of homotopy theory without using topological
spaces. Instead, it relies on certain diagrams of sets. Homology can be described elegantly in this theory; in fact,
it essentially amounts to taking the free abelian group on the simplicial set.
This topic has been taken.

The fundamental groupoid. The fundamental group of a topological space depends on the choice of a basepoint,
which can be inconvenient. The fundamental groupoid is something like the fundamental group, but does not depend on
any choice. However, it is a category instead of a group. This can be extended to higher homotopy groups through
the use of higher categories.

The derived category. The Kunneth formula states that H•(X × Y)=H•(X)
⊗ H•(Y), up to some error terms involving Tor's. Similarly, the universal coefficient theorem
states that H•(X; R)=H•(X; Z) ⊗ R, again up to some error terms
involving Tor's. Derived categories provide a way to eliminate these error terms, which is very desirable.
The basic idea of derived categories is to systematically work with the complexes that compute homology, rather than
homology itself. Derived categories are very important in a wide variety of subjects these days.

Classifying spaces

Eilenberg–MacLane spaces. Let G be a group and let n≥1 be an integer; if n>1 then assume that
G is commutative. One can then show that there is a unique homotopy type X with πn(X)=G and
πk(X)=1 for k≠n. This homotopy type is usually denoted K(G, n) and called an Eilenberg–MacLane
space. One reason that these spaces are interesting is that they represent cohomology: giving an element of
Hn(X; G) is the same as giving a homotopy class of maps X → K(G, n).

Group cohomology. Let G be a group. As stated above, there is an associated homotopy type K(G, 1). The
(co)homology of K(G, 1) is an invariant of the group G, called group (co)homology. One can give a purely algebraic
construction of group (co)homology, however, the topological perspective is often useful as well.

The classifying space for vector bundles. Let n≥1 be an integer. There is then a space B with the
following property: homotopy classes of maps from an arbitrary space X into B are canonically in bijection with
complex vector bundles of rank n on X. The space B is infinite dimensional, but can be described
explicitly as a Grassmannian. Using classifying spaces, one can associate cohomology classes to vector bundles;
this is the theory of characteristic classes.

Higher homotopy groups

The long exact sequence. A fibration is the analogue in the world of homotopy theory to the concept of
a short exact sequence. Given a fibration F → X → B, there is a long exact sequence relating the
homotopy groups of F, X and B. This can be used to calculate some higher homotopy groups.
This topic has been taken.

The Freudenthal suspension theorem. This result states that, in certain cases, homotopy groups of
suspensions of a space stabilize; precisely, πn+k(SkX) is independent of k when k is large.
This leads to the concept stable homotopy groups, and a whole stable homotopy theory.

The Hurewicz homomorphism. There is a natural map from homotopy to singular homology, called the
Hurewicz homomorphism. Hurewicz showed that for a simply connected space, the first non-zero homotopy and homology
groups are isomorphic via this map.
This topic has been taken.

Other topics

Spectral sequences. As mentioned above, associated to a fibration of spaces is a long exact sequence
of homotopy groups. The behvior of homology in a fibration is not as simple — it is described by an algebraic
construct called a spectral sequence. One can use spectral sequences to compute the homology of certain spaces,
such as Lie groups and some loop spaces.

Steenrod squares. Steenrod constructed a natural map Hn(X; F2) →
Hn+i(X F2) for each i, which is now called the Steenrod square Sqi.
This gives the cohomology of a space with F2 coefficients extra structure, and allows one
to distinguish spaces using cohomology that one could not before.

Morse theory. This theory gives a way to decompose a manifold using critical points of functions on it.
John Milnor has a nice book on the subject.
This topic has been taken.

The Gauss–Bonnet theorem. Given a surface in Euclidean space (or, more abstractly, a Riemannian
manifold of dimension 2), one can speak of the curvature of the surface at a point. The Gauss–Bonnet theorem
states that the total curvature (the integral over the surface of the curvature at each point) is equal to the Euler
characteristic of the surface. Like the de Rham theorem mentioned above, this is remarkable since it shows that the
total curvature of a surface is a topological invariant, even though this is not apparent in its definition.
This topic has been taken.